You'll need to go through the normal gambit of troubleshooting network issues. Like try a different cable or a different port on the router. Also, try checking your router for a firmware updates. There have been issues with past firmware with the Slingbox obtaining DHCP address which were typically resolved with a router firmware update. I know in particularly, that these problems were carried on to the Linux router kernels like DD-WRT, FreeWRT, OpenWRT and Totato. In which case, usually reverting to the original manufacturer firmware allowed the DHCP work long enough to assign a static IP address on the Slingbox.
As I understand it is not entirely uncommon for malfunctioning AC adapter to cause the symptoms that you've described. On the bottom of the Slingbox, you should see the power requirements; voltage, amps, and polarity. Make sure the AC adapter connected to your Slingbox has the same output sepecifications. Otherwuse, if you have a compatible power supply then you can test it with your Slingbox to see if there's any change.